Question

If alfa and bita are the zeros of polynomial 2x^2-5x+7 find a polynomial whose zeros are 2alfa+3 and 2bita+3

if u can't understand there is also a picture attached

Answer 1

Hi friend!!!

given œ,ß are the roots of 2x²-5x+7

we know that œ+ß=-b/a=5/2

and ϧ=c/a=7/2

Now , the quadratic polynomial whose zeros are 2œ+3,2ß+3=??

Now, sum of zeros =2œ+2ß+6=2(œ+ß)+6=2(5/2)+6=5+6=11

product of zeros = (2œ+3)(2ß+3)

=4œß+6œ+6ß+9

=4(7/2)+6(5/2)+9

=14+15+9

=38

Required quadratic polynomial = x²-(sum of zeros )x+product of zeros

x²-11x+38 is the required polynomial.



I hope this will help u ;)

Answer 2

when p and q are the zeroes of the polynomial, then the polynomial formed will be
x²-(p+q)x +pq

Let me assume alpha = a and beta = b.
The given polynomial is 2x²-5x+7.
a+b = 5/2
ab = 7/2

Now, we have to form a polynomial whose zeroes are 2a+3 and 2b+3.
So, the polynomial formed is:

x²-(sum of zeroes)x + (product of zeroes)
x² -(2a+3+2b+3)x +(2a+3)(2b+3)
x² -(2(a+b)+6)x + 9 +3(2a +2b)+ 4ab
x² -(2(a+b)+6)x + 9 +6(a +b)+ 4ab

Now, just substitute the values of a+b and ab, and you will get the polynomial.

x² -(2(5/2)+6)x + 9 +6(5/2)+ 4(7/2)

x² -(5+6)x + 9 +3(5)+ 2(7)

x²-11x+9+15+14

x²-11x+38

Hence , This is the required polynomial.

I hope you understand the approach.
All the best!